<h2>题目编号 : 153</h2>
<div style="color:#666;font-size:80%;">05 May 2007</div><br />
<div class="problem_content">
<p>As we all know the equation <var>x</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=-1 has no solutions for real <var>x</var>.
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If we however introduce the imaginary number <var>i</var> this equation has two solutions: <var>x=i</var> and <var>x=-i</var>.
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If we go a step further the equation (<var>x</var>-3)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=-4 has two complex solutions: <var>x</var>=3+2<var>i</var> and <var>x</var>=3-2<var>i</var>.
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<var>x</var>=3+2<var>i</var> and <var>x</var>=3-2<var>i</var> are called each others' complex conjugate.
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Numbers of the form <var>a</var>+<var>bi</var> are called complex numbers.
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In general <var>a</var>+<var>bi</var> and <var>a</var><img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /><var>bi</var> are each other's complex conjugate.</p>
<p>A Gaussian Integer is a complex number <var>a</var>+<var>bi</var> such that both <var>a</var> and <var>b</var> are integers.
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The regular integers are also Gaussian integers (with <var>b</var>=0).
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To distinguish them from Gaussian integers with <var>b</var> <img src='images/symbol_ne.gif' width='11' height='10' alt='&ne;' border='0' style='vertical-align:middle;' /> 0 we call such integers &quot;rational integers.&quot;
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A Gaussian integer is called a divisor of a rational integer <var>n</var> if the result is also a Gaussian integer.
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If for example we divide 5 by 1+2<var>i</var> we can simplify <img src="project/images/p_153_formule1.gif" border="0" style="vertical-align:middle" alt="" /> in the following manner:
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Multiply numerator and denominator by the complex conjugate of 1+2<var>i</var>: 1<img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' />2<var>i</var>.
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The result is 
<img src="project/images/p_153_formule2.gif" border="0" alt="" style="vertical-align:middle;" />.
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So 1+2<var>i</var> is a divisor of 5.
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Note that 1+<var>i</var> is not a divisor of 5 because <img src="project/images/p_153_formule5.gif" border="0" style="vertical-align:middle;" alt="" />.
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Note also that if the Gaussian Integer (<var>a</var>+<var>bi</var>) is a divisor of a rational integer <var>n</var>, then its complex conjugate (<var>a</var><img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /><var>bi</var>) is also a divisor of <var>n</var>.</p>
<p>In fact, 5 has six divisors such that the real part is positive: {1, 1 + 2<var>i</var>, 1 <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> 2<var>i</var>, 2 + <var>i</var>, 2 <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> <var>i</var>, 5}.
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The following is a table of all of the divisors for the first five positive rational integers:</p>
<table align="center" border="1">
<tr><td width="20">
<var>n</var></td><td> Gaussian integer divisors<br />
with positive real part</td><td>Sum s(<var>n</var>) of <br />these

divisors</td></tr><tr>
<td>1</td><td>1</td><td>1</td>
</tr><tr>
<td>2</td><td>1, 1+<var>i</var>, 1-<var>i</var>, 2</td><td>5</td>
</tr><tr>
<td>3</td><td>1, 3</td><td>4</td>
</tr><tr>
<td>4</td><td>1, 1+<var>i</var>, 1-<var>i</var>, 2, 2+2<var>i</var>, 2-2<var>i</var>,4</td><td>13</td>
</tr><tr>
<td>5</td><td>1, 1+2<var>i</var>, 1-2<var>i</var>, 2+<var>i</var>, 2-<var>i</var>, 5</td><td>12</td>
</tr></table>
<p>For divisors with positive real parts, then, we have: <img src="project/images/p_153_formule6.gif" border="0" style="vertical-align:middle" alt="" />.</p>
<p>For 1 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 10<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />, <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /> s(<var>n</var>)=17924657155.</p>
<p>What is <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /> s(<var>n</var>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 10<img src="" style="display:none;" alt="^(" /><sup>8</sup><img src="" style="display:none;" alt=")" />?</p>
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